What is the Range of a Rational Function?

[FlO-rida]

how can i find the range of a rational function

for ex. y=1/x+1

 

[jbunniii]

What's the definition of the range of a function?

 

[Angry Citizen]

Think what "domain" and "range" are. What can "x" be, and what can "x" not be? With that in mind, what can "y" be and what can "y" not be?

 

[Mark44]

Also, what exactly is your function? Most would interpret what you have written like so:
$$y = \frac{1}{x} + 1$$

I suspect that what you really meant was this:
$$y = \frac{1}{x + 1}$$

When you write a fraction on a single line, use parentheses. The second version above should be written this way: y = 1/(x + 1)

 

[FlO-rida]

ok how would we find the range for that

 

[Angry Citizen]

Do you know what domain and range are? Your instructor would not be giving you rational functions without a thorough treatment of the concepts of domain and range, and how they relate to rational functions.

 

[Mark44]

ok how would we find the range for that

The range for what? As I already said in post 4, it's not clear what you're working with.

 

[FlO-rida]

the range for 1/(x+1) .the thing is that i am doing an assignment on rational functions that is ment to be completed without help from the teacher (it will be explained upon completion). i have already researched horizontal and vertical asymptote of rational functions, as well as the domain but i still can't find anything on how to express the range (not in interval notation). don't get me wrong i know what range is.

 

[jbunniii]

the range for 1/(x+1) .the thing is that i am doing an assignment on rational functions that is ment to be completed without help from the teacher (it will be explained upon completion). i have already researched horizontal and vertical asymptote of rational functions, as well as the domain but i still can't find anything on how to express the range (not in interval notation). don't get me wrong i know what range is.

[EDIT] Weird, it was all garbled on my screen when I responded, but now it looks fine, both in my quoted version and in the original post!

OK, so the function of interest is

$$y = \frac{1}{x+1}$$

What's the domain of this function?

 

[FlO-rida]

thats wat i mean i just didnt know how to do it

 

[FlO-rida]

y=\\frac{1}{x+3}

 

[jbunniii]

You can click on my equation (or any typeset equation on this site) and it will give you a pop-up window with the Latex code that produced it.

 

[FlO-rida]

[x]\neq[/-1]

 

[FlO-rida]

i just made an acc. yesterday

 

[jbunniii]

OK, good. So let's consider two cases:

$$x < -1$$

$$x > -1$$

Start by focusing on the first case, so we're just considering $x < -1$. For $x$ in this range, can I make the function as big as I like? Can I make it as small as I like? If not, then what are some bounds? (Even if they're not the tightest possible bounds, it's a start.)

 

[FlO-rida]

it would go on to infinity, but wouldn't the range be expressed using y values

 

[jbunniii]

it would go on to infinity, but wouldn't the range be expressed using y values

Yes, the range is expressed using y values.

What do you mean by "it would go on to infinity"? Can you make it infinitely large (positive)? Can you make it infinitely small (negative)?

If $x < -1$, then can y be positive at all?

 

[FlO-rida]

infinitely small

 

[FlO-rida]

so how would you express the range ({y|y...)

 

[jbunniii]

infinitely small

OK, so you can make y as negative as you like by varying x over the interval $(-\infty, -1)$. How large can you make y if x is in the same interval?

 

[FlO-rida]

sorry i am still in math 20 and we didnt take interval notation yet

 

[jbunniii]

sorry i am still in math 20 and we didnt take interval notation yet

No problem. It's just another way of writing

$$x < -1$$

So how big can y get if x is in this interval? Can you make y be positive? Can you make y be zero?

 

[FlO-rida]

i think you can make it zero, but can you please explain

 

[jbunniii]

zero

What value of x causes y = 0?

 

[FlO-rida]

infinity? sorry but i just don't get this

 

[FlO-rida]

i got that from punching in random x values on my graphing calc. (10, 20, 30) and as they got bigger the y values got smaller but never reached zero.

 

[jbunniii]

i got that from punching in random x values on my graphing calc. (10, 20, 30) and as they got bigger the y values got smaller but never reached zero.

That's right. You can make y as close to 0 as you like, but it never quite gets there. You can also make y as negative as you like.

So that means that for $x < -1$, the range of possible y values is $y < 0$.

Other ways of expressing the same thing are

$$\{y | y < 0\}$$

or

$$(-\infty,0)$$

Now let's consider the other half of the domain: $x > -1$. What range of y values are possible here?

 

[FlO-rida]

positve infinity or y > 0

 

[jbunniii]

positve infinity or y > 0

OK, so if $x > 1$ then the possible range of y values is $y > 0$.

Now put the two halves together to get the total range of the function.

 

[HallsofIvy]

The "natural domain" of a function where you are given a formula for it is just the set of all x values to which the formula can be applied. Here, the only arithmetic operation is "divide by x+ 1". You can divide any number except 0 so you can do that calculation for any number except x= -1. The domain is "all x except -1".

In interval notation that would be $(-\infty, -1)\cup (-1, \infty)$.

The "range" is the set of possible y values.

One way of finding the range is to try to invert the function. If y= 1/(x+1) then x+1= 1/y and x= (1/y)- 1. Since we can divide by any number except 0, y can take any value except 0. The range is "all y except 0".

For this simple function, you could also have thought, since a/b= 0 gives immediately a= 0 by multiplying both sides by b, "a fraction is 0 if and only if its numerator is 0". Here the numerator is the constant 1 which is never 0. y can never be 0.

 

[FlO-rida]

Thanks for the help, and just to be sure, the range is always expressed as y $$\neq$$ ...

 

[Char. Limit]

No, only sometimes, like here.